Asymptotic expansions uniqueness

Using field theoretic methods this divergence has been analyzed to some extent. The analysis strongly suggests that the expansion is in fact an asymptotic expansion in the mathematical sense. Using the powerful method of Borel transformation one can resum the expansion to get a unique finite result. We can thus state that the theory is well defined and all the qualitative... 

A very important property of an asymptotic expansion is the manner in which it converges to the function that it is intended to represent. Two facts can be stated that relate intimately to the nature of the convergence of an asymptotic expansion. First, if a function such as T(x e) has an asymptotic expansion for small e (either for all x or at least in some subdomain of x), then this expansion is unique (at least in the subdomain). However, second, more than one function T may have the same asymptotic representation through any finite number of terms. The second of these statements implies that one cannot sum an asymptotic expansion to find a unique function T(x e) as would be possible (in the domain of convergence) with a normal power-series representation of a function. This distinction between an asymptotic and infinite-series representation is reflected in a more formal statement of the convergence properties of both an infinite series and an asymptotic expansion. In the case of an infinite-series representation of some function T(x e), namely,... 

The most straightforward way to evaluate corrections to the large-dimension limit is to compute a 1/D expansion [5]. The energy eigenvalues each have a unique asymptotic expansion in the form... 

A function y can have many asymptotic expansions simply because there are many sets of asymptotic sequences 8 that could be selected. However, for a given asymptotic sequence, the asymptotic expansion is unique, and the coefficients yj are determined as follows. First divide Eq. 6.37 by Sq to see... 

However, MET is not a unique theory accounting for the higher-order concentration corrections. Similar results were obtained within the fully renormalized YLS (Yang-Lee-Shin) theory [132], which is also integrodiffer-ential and employs the kernels containing concentration corrections as compared to those in IET. It was shown in Ref. 41 that both these theories, MET and YLS, provide the correct asymptotic expressions for binary kinetics, but differ slightly in the nonlinear terms of the concentration expansion. There were also a number of other attempts to overcome the concentration limitations of the theory made by the same Korean group earlier (in superposition approximation [139,141,142]) and later [numerous multiparticle kernel (MPK) theories [51,126]]. 

The objective of this note is to make this last statement mathematically more precise and in the course of the analysis delineate and solve problems which do not appear in the classical treatment. In trying to examine the morphological stability of planar fronts one must first find a special planar solution and show it is marginally stable among all planar solutions before one can use it as the base of a perturbation expansion. In section 2 we examine the existence, uniqueness and asymptotic behavior of... 

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